Here's an analogy. Suppose I have a theater with 2 seats in the first row, 8 in the second, 18 in the third, 32 in the fourth, etc. The tickets for the first row cost $20, the second row $50, the third row $75, the fourth $125, and so on. (I know, it is a weird theater, but this is just an analogy.) Spectators are forbidden to sit in someone else's lap or stand in the aisles. If 12 people show up for the performance, they first occupy the 2 seats of the first row, the next 8 patrons occupy the second row, and the last 2 spectators sit somewhere in the third row. Further back in the theater, the rows have more seats, but they are empty. If a patron wants to change rows – from row 3 to row 4, for example – he has to pay the extra $50: the difference in the price of the tickets in the different rows. If he moves the other way, from row 4 to row 3, he is reimbursed the $50. Now, if a wealthy person in row 1 wants to move to row 4, he will be required to pay $105. That is, money must be paid or received to move from one row to another. All these changes assume that the seat someone desires is unoccupied. If the group of spectators is short of money (no energy), they will occupy the seats of the first rows as long as there are seats available. If the group is wealthy (has lots of energy), they can jump from one seat to another as long as they have enough money (energy) to afford the higher prices. The amount they have to pay depends only on the difference in the price of the seats in each row. End of analogy.
At 0 K, absolute temperature (−273 °C), there is no energy whatsoever, so all the electrons occupy the lowest allowed energy levels. At room temperature, 300 °C, there is quite a large amount of thermal energy, and electrons start moving from one level to another, leaving empty seats that can be occupied by other electrons, absorbing or emitting photons as they move.
Figure 1.14 shows the transitions observed in the hydrogen atom. The groups of lines were named later by those who found them.
Have you ever wondered why, when we walk on the second floor, we do not fall through it and land on the first floor? Think about it. The typical size of an atom is 5 × 10−10 m, and the size of a nucleus is about 30 000 times smaller, 1.6 × 10−15 m. All the mass is concentrated in the nucleus. The atoms are, for all practical purposes, composed of empty space. So why does the empty space of my shoes do not go through the empty space of the tiles on the second floor? It is not due to electrostatic repulsion. Both the soles of my shoes and the tiles are electrically neutral. The reason we do not fall through the floor is the Pauli exclusion principle. The electrons in the sole cannot find a lower energy level on the atoms of the tile. The Pauli exclusion principle not only keeps us safe on the second floor but also explains why material physical objects have any volume at all. It also explains friction. The atoms of the sole locate themselves in a preferential position with the atoms of the floor, and they resist moving. How intense the friction is depends on the crystallographic structure of the two surfaces (Emily Conover, “Giving Friction the Slip”, Science News, 3 August 2019).
Figure 1.14 The observed energy lines of the hydrogen atom corresponding to all the transitions between different atomic levels.
In the next chapter, I discuss how these single unique energy levels that Bohr postulated explain the electric properties of different materials.
1.9 Summary and Conclusions
Perhaps the best way to summarize what we have covered in this chapter is to take a look at Figure 1.15. Observations of the sun's light spectrum and the spectra of different gases with their distinct lines resulted in heuristic relationships that relate the frequency of the missing lines to an expression consisting of just a constant and integer numbers. Einstein, working with the photoelectric effect, postulated the dual nature of light acting as both a wave and a particle, which we now call the photon.
Figure 1.15 The scientific and experimental work that led to the Bohr planetary model of the atom.
On the atomic side, Mendeleev classified the known elements by their weight, and, in the process, he left some empty spaces to add future elements. Thomson determined that electrons are tiny negatively charged particles and Rutherford, with his alpha ray measurements, concluded that the nucleus is concentrated in a very small region at the center of the atom. Millikan was able to measure the charge of electrons.
Based on previous theoretical and experimental work, Bohr proposed his planetary model of the atom with discrete and stable energy levels. His model included all the developments of atomic theory known to that date and explained the previous optical observations and measurements beautifully by considering how electrons move from one level to another by accepting or releasing packets of energy.
If you are comfortable with these conclusions, you are ready to go to the next chapter. You may peruse the three following appendices for a few more details.
Appendix 1.1 Some Details of the Bohr Model
Four quantum numbers uniquely define an electron:
The principal quantum number, n, defines the orbits and, therefore, the energy of the electrons. The energy released or absorbed as the electrons change orbits is determined exclusively by the value of n. The allowed values of n are any positive integers: 1, 2, 3, 4, etc.
Electrons have two spins: up and down.
Electrons also have angular momentum. The angular quantum number, ℓ, is associated with the shape of the orbits. The value of ℓ is also an integer number between 0 and n.
The magnetic quantum number, ml, is associated with the orientation of the orbits. It is also an integer number between –ℓ and +ℓ.
Electrons follow these rules:
The electrons in the first orbit, n = 1, can have two spins, but both ℓ and ml are 0. Thus the first orbit can hold only two electrons.
The electrons in the second orbit, n = 2, can have two spins and two angular quantum numbers (ℓ equal to 0 or 1). Associated with ℓ = 0, only one ml value is possible, ml = 0. But for ℓ = 1, there are three possible values of ml: −1, 0, and +1. So, the total number of electrons in the second orbit is eight: that is, four ml times two spins each.
You can do the calculations and prove that the third orbit can hold up to 18 electrons, and so on.
Figure 1.16 shows one way of visualizing these levels, including the relative energy of the orbits of the Bohr atom and the order in which the orbits are filled. At 0 K absolute temperature, the electrons first fill up the 1s band (two electrons), the next two electrons reside in the 2s band, and the next six are in the 2p band, etc., climbing up the energy level stairs. Note that level 4s fills up before 3d. (By the way, the letters mean s for sharp, p for principle, d for diffuse, and f for fundamental.)
Another point you may wonder about is why we write the 3d level at higher energy